Conservation laws of scaling-invariant field equations

TL;DR

This paper addresses systematic generation of local conservation laws for scaling-invariant PDEs by introducing a simple algebraic formula $\Phi^{\alpha}_{\omega}$ built from adjoint-symmetries and the scaling symmetry, bypassing the adjoint invariance condition and homotopy methods. It proves a scaling-weight relation $\Phi^{\alpha}_{Q} \simeq w_{Q} \Psi^{\alpha}_{Q}$ (Theorem 2.2) and shows that all noncritical conservation laws are obtainable from adjoint-symmetries (Corollary 2.4). The authors demonstrate explicit recursions and densities for KdV and sine-Gordon, extend to vector mKdV, and apply the framework to Euler fluids, Yang-Mills, and Einstein gravity, including nonlocal structures. Overall, the work provides a unifying, algebraic mechanism to generate and classify conservation laws across a wide range of physically important systems, with potential to reveal nonlocal conservation laws from nonlocal adjoint-symmetries.

Abstract

A simple conservation law formula for field equations with a scaling symmetry is presented. The formula uses adjoint-symmetries of the given field equation and directly generates all local conservation laws for any conserved quantities having non-zero scaling weight. Applications to several soliton equations, fluid flow and nonlinear wave equations, Yang-Mills equations and the Einstein gravitational field equations are considered.